OFFSET
0,2
COMMENTS
These are the numbers d(n,3) studied by J. L. Martin. - N. J. A. Sloane, Sep 13 2014
For n >= 0, the number of ternary sequences of length n+1, that contain at least one pair of same consecutive digits. - Armend Shabani, Apr 10 2019
LINKS
Philippe Flajolet and Robert Sedgewick, Combinatorial Parameters and MGFs, lecture slides Analytic Combinatorics, 2012.
J. L. Martin, The slopes determined by n points in the plane [Dead link]
Jeremy L. Martin, The slopes determined by n points in the plane, arXiv:math/0302106 [math.AG], 2003-2006; Duke Math. J. 131 (2006), no. 1, 119-165. See table of d(n,k), but beware errors.
Index entries for linear recurrences with constant coefficients, signature (5,-6).
FORMULA
E.g.f.: 3*exp(3x) - 3*exp(2x).
See Mathematica code for a more transparent version of the e.g.f.
Generally for an m-ary word of length n: m*exp(m*x) - m*exp((m-1)*x)
From Alois P. Heinz, Jan 20 2013: (Start)
a(n) = 3*(3^n-2^n) = 3*A001047(n).
G.f.: 3*x/((3*x-1)*(2*x-1)).
(End)
a(n) = A217764(n,5). - Ross La Haye, Mar 27 2013
EXAMPLE
a(2) = 15 because the length 2 words on alphabet {0,1,2} are: 00, 01, 02, 10, 11, 12, 20, 21, 22 and we sum respectively 1 + 2 + 2 + 2 + 1 + 2 + 2 + 2 + 1 = 15.
MAPLE
a:= n-> 3*(3^n-2^n):
seq(a(n), n=0..30); # Alois P. Heinz, Jan 20 2013
MATHEMATICA
nn=28; Range[0, nn]!CoefficientList[Series[D[(1+y(Exp[x]-1))^3, y]/.y->1, {x, 0, nn}], x]
(* Second program: *)
LinearRecurrence[{5, -6}, {0, 3}, 30] (* Jean-François Alcover, Jan 09 2019 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Geoffrey Critzer, Jan 20 2013
STATUS
approved